Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates

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DE LA FACULTÉ DES SCIENCES

Mathématiques

JEANDOLBEAULT ANDMICHAŁKOWALCZYK

Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates

Tome XXVI, n^o4 (2017), p. 949-977.

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Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates

Jean Dolbeault⁽¹⁾ and Michał Kowalczyk⁽²⁾

ABSTRACT. — This paper is devoted to the Lin–Ni conjecture for a semi-linear elliptic equation with a super-linear, sub-critical nonlinearity and homogeneous Neumann boundary conditions. We establish a new rigidity result, that is, we prove that the unique positive solution is a constant if the parameter of the problem is below an explicit bound that we relate with an optimal constant for a Gagliardo–Nirenberg–Sobolev interpolation inequality and also with an optimal Keller–Lieb–Thirring inequality. Our results are valid in a sub-linear regime as well. The rigidity bound is obtained by nonlinear flow methods inspired by recent results on compact manifolds, which unify nonlinear elliptic techniques and the carré du champmethod in semi-group theory. Our method requires the convexity of the domain. It relies on integral quantities, takes into account spectral estimates and provides improved functional inequalities.

RÉSUMÉ. — Cet article est consacré à la conjecture de Lin–Ni pour une équation semi-linéaire elliptique avec non-linéarité super-linéaire, sous- critique et des conditions de Neumann homogènes. Nous établissons un résultat de rigidité, c’est-à-dire nous prouvons que la seule solution posi- tive est constante si le paramètre du problème est en dessous d’une borne

Keywords:semilinear elliptic equations, Lin–Ni conjecture, Sobolev inequality, inter- polation, Gagliardo–Nirenberg inequalities, Keller–Lieb–Thirring inequality, optimal con- stants, rigidity results, uniqueness,carré du champmethod, CD(ρ,N) condition, bifurca- tion, multiplicity, generalized entropy methods, heat flow, nonlinear diffusion, spectral gap inequality, Poincaré inequality, improved inequalities, non-Lipschitz nonlinearity, compact support principle.

2010Mathematics Subject Classification:35J60, 26D10, 46E35.

(1) Ceremade, UMR CNRS n^◦7534, Université Paris-Dauphine, PSL research university, Pl. de Lattre de Tassigny, 75775 Paris Cédex 16, France —

dolbeaul@ceremade.dauphine.fr

(2) Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (CNRS UMI n^◦2807), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile — kowalczy@dim.uchile.cl

J.D. thanks the LabEx CIMI, the ANR projects STAB, NoNAP and Kibord for support, and the MathAmSud project QUESP. Both authors thank the ECOS (Chile-France) project C11E07. M.K. thanks Chilean grants Fondecyt 1130126 and Fondo Basal CMM-Chile. Both authors are indebted to an anonymous referee for his careful reading of the paper and useful suggestions.

explicite, reliée à la constante optimale d’une inégalité d’interpolation de Gagliardo–Nirenberg–Sobolev et aussi à une inégalité de Keller–Lieb–

Thirring optimale. Nos résultats sont également valides dans un régime sous-linéaire. La borne de rigidité est obtenue par des méthodes de flots non-linéaires inspirées de résultats récents sur les variétés compactes, qui unifient des techniques d’équations elliptiques non-linéaires et la méthode du carré du champ en théorie des semi-groupes. Notre méthode requiert la convexité du domaine. Elle repose sur des quantités intégrales, prend en compte des estimations spectrales et fournit des inégalités améliorées.

1. Introduction and main results

Let us assume that Ω is a bounded domain inR^dwith smooth boundary.

To avoid normalization issues, we shall assume throughout this paper that

|Ω|= 1.

The unit outgoing normal vector at the boundary is denoted bynand∂nu=

∇u·n. We shall denote by 2^∗ = _d−2^2d the critical exponent ifd>3 and let 2^∗ =∞ifd= 1 or 2. Assume first thatpis in the range 1< p <2^∗−1 = (d+ 2)/(d−2) ifd >3, 1< p < ∞if d= 1 or 2, and let us consider the three following problems.

(P1) For which values ofλ >0 does the equation

−∆u+λ u=u^p in Ω, ∂nu= 0 on ∂Ω (1.1) has a unique positive solution?

(P2) For anyλ >0, let us define µ(λ) := inf

u∈H¹(Ω)\{0}

k∇uk²_L2(Ω)+λkuk²_L2(Ω)

kuk²_Lp+1(Ω)

. For which values ofλ >0 do we haveµ(λ) =λ?

(P3) Assume thatφis nonnegative function in L^q(Ω) withq= ^p+1_p−1 and denote byλ₁(Ω,−φ) the lowest eigenvalue of the Schrödinger oper- ator−∆−φ. Let us consider the optimal inequality

λ₁(Ω,−φ)>−ν kφkL^q(Ω)

∀φ∈L^q₊(Ω).

For which values ofµdo we know thatν(µ) =µ?

The three problems are related. Uniqueness in (P1) means thatu=λ^1/(p−1) while equality casesµ(λ) =λin (P2) andν(µ) =µand (P3) are achieved by constant functions and constant potentials respectively. We define a thresh- old value µi withi = 1, 2, 3 such that the answer to (Pi) is yes ifµ < µi

and no ifµ > µi.

Our method is not limited to the casep >1. Ifpis in the range 0< p <1, the three problems can be reformulated as follows.

(P1) For which values ofλ >0 does the equation

−∆u+u^p=λ u in Ω, ∂nu= 0 on ∂Ω (1.2) have a unique nonnegative solution?

(P2) For anyµ >0, let us define λ(µ) := inf

u∈H¹(Ω)\{0}

k∇uk²_L2(Ω)+µkuk²_Lp+1(Ω)

kuk²_L2(Ω)

. For which values ofµ >0 do we haveλ(µ) =µ?

(P3) Assume thatφis a nonnegative function in L^q(Ω) withq=^1+p_1−p and still denote by λ₁(Ω, φ) the lowest eigenvalue of the Schrödinger operator−∆ +φ. Let us consider the optimal inequality

λ1(Ω, φ)>ν(kφ⁻¹k_Lq(Ω)) ∀φ∈L^q₊(Ω).

For which values ofµdo we know thatν(µ) =µ?

The problems of the range 0< p <1 and 1< p <2^∗ can be unified. Let us define

ε(p) = p−1

|p−1|

and observe that kukL^p+1(Ω) 6 kukL²(Ω) if p < 1, kukL²(Ω) 6 kukL^p+1(Ω)

if p > 1, so that ε(p) kuk_Lp+1(Ω) − kuk_L2(Ω)

is nonnegative. Our three problems can be reformulated in terms of estimating bounds,µiwithi= 1, 2, 3, defined as follows.

(P1) Let us consider the equation

−ε(p) ∆u+λ u−u^p= 0 in Ω, ∂_nu= 0 on ∂Ω (1.3) and define

µ₁:= inf{λ >0 : (1.3) has a non constant positive solution}.

We shall say thatrigidityholds in (1.3) ifu=λ^1/(p−1)is its unique positive solution.

(P2) For anyµ > 0, takeλ(µ) as the best (i.e. the smallest if ε(p)>0 and the largest ifε(p)<0) constant in the inequality

k∇uk²_L2(Ω)>ε(p)h

µkuk²_Lp+1(Ω)−λ(µ)kuk²_L2(Ω)

i ∀u∈H¹(Ω). (1.4) Here we denote byµ7→λ(µ) the inverse function ofλ7→µ(λ). Let

µ2:= inf{λ >0 : µ(λ)6=λin (1.4)}.

(P3) Let us consider the optimalKeller–Lieb–Thirring inequality ν(µ) =−ε(p) inf

φ∈Aµ

λ₁(Ω,−ε(p)φ) (1.5) where the admissible set for the potentialφis defined by

Aµ:=n

φ∈L^q₊(Ω) : kφ^ε(p)k_Lq(Ω)=µo andq= (p+ 1)/|p−1|. Let

µ₃:= inf{µ >0 : ν(µ)6=µin (1.5)}.

Finally let us define Λ? as thebest constant in the interpolation inequality k∇uk²_L2(Ω)> Λ?

p−1

hkuk²_Lp+1(Ω)− kuk²_L2(Ω)

i ∀u∈H¹(Ω). (1.6) Let us observe that Λ_?may depend on p.

Theorem 1.1. — Assume that d >2, p∈(0,1)∪(1,2^∗−1) andΩ is a bounded domain inR^d with smooth boundary such that|Ω|= 1. With the above notation, we have

0< µ16µ2=µ3= Λ?

|p−1|

and, withλ(µ)andν(µ)defined as in(1.4)and(1.5), the following properties hold:

(P1) Rigidity holds in(1.3)for any λ∈(0, µ1).

(P2) The functionµ7→λ(µ)is monotone increasing, concave ifp∈(0,1), convex ifp∈(1,2^∗−1)andλ(µ) =µif and only ifµ6µ2. (P3) For anyµ >0,ν(µ) =λ(µ).

This result is inspired by a series of recent papers on interpolation inequal- ities, rigidity results and Keller–Lieb–Thirring estimates on compact mani- folds. Concerning Keller–Lieb–Thirring inequalities, we refer to [16, 17, 20], and to the initial paper [31] by J.B. Keller whose results were later redis- covered by E.H. Lieb and W. Thirring in [35]. For interpolation inequalities on compact manifolds, we refer to [15, 19, 23] and references therein. In our case, the absence of curvature and the presence of a boundary induce a num- ber of changes compared to these papers, that we shall study next. Beyond the properties of Theorem 1.1 which are not very difficult to prove, our main goal is to get explicit estimates ofµ_i and Λ?.

Let us define

λ₂:=λ₂(Ω,0)

which is the second (and first positive) eigenvalue of−∆ on Ω, with homo- geneous Neumann boundary conditions. Recall that the lowest eigenvalue

of−∆ isλ₁ = 0 and that the corresponding eigenspace is spanned by the constants. For this reasonλ₂is often called thespectral gapand the Poincaré inequality sometimes appears in the literature as thespectral gap inequality.

Finally let us introduce the number

θ_?(p, d) = (d−1)²p

d(d+ 2) +p. (1.7)

Theorem 1.2. — Assume that d>1 andΩis a bounded domain inR^d with smooth boundary such that|Ω|= 1. With the above notation, we get the following estimates:

1−θ_?(p, d)

|p−1| λ26µ16µ2=µ3= Λ?

|p−1| 6 λ₂

|p−1|

for any p ∈ (0,1)∪(1,2^∗−1). The lower estimate holds only under the additional assumptions thatΩis convex andd>2.

Before giving a brief overview of the literature related to our results, let us emphasize two points. We first notice that limp→(d+2)/(d−2)θ?(p, d) = 1 if d>3 so that the lower estimate goes to 0 as the exponentpapproaches the critical exponent. This is consistent with the previously known results on rigidity, that are based on Morrey’s scheme and deteriorate aspapproaches (d+ 2)/(d−2). In the critical case, multiplicity may hold for any value ofλ, so that one cannot expect that rigidity could hold without an additional assumption. The second remark is the fact that the convexity of Ω is essential for known results in the critical case and one should not be surprised to see this condition also in the sub-critical range. This assumption is however not required in the result of C.-S. Lin, W.-M. Ni, and I. Takagi in [37]. Compared to their paper, what we gain here when p > 1 is a fully explicit estimate which relies on a simple computation. The case p < 1 has apparently not been studied yet.

It is remarkable that the casep= 1 is the endpoint of the two admissible intervals inp. We may notice that

θ?(1, d) = (d−1)² (d+ 1)²

is in the interval (0,1) for any d>2. The casep= 1 is a limit case, which corresponds to the logarithmic Sobolev inequality

k∇uk²_L2(Ω)−Λ?

2 Z

Ω

|u|² log |u|² kuk²_L2(Ω)

!

dx>0 ∀u∈H¹(Ω) (1.8) where Λ?denotes the optimal constant, and by passing to the limit asp→1 in (1.6), we have the following result.

Corollary 1.3. — If d > 2 and Ω is a bounded domain in R^d with smooth boundary such that|Ω|= 1, then

4d

(d+ 1)²λ26Λ?6λ2.

It is also possible to define a family of logarithmic Sobolev inequalities depending on λ, or to get a parametrized Keller–Lieb–Thirring inequality and find thatλ= Λ_? corresponds to a threshold value between a linear de- pendence of the optimal constant inλand a regime in which this dependence is given by a strictly convex function ofλ. The interested reader is invited to consult [16, Corollaries 13 and 14] for similar results on the sphere.

The existence of a non-trivial solution to (1.1) bifurcating from the con- stant ones forλ=λ₂/(p−1) has been established for instance in [36] when p >1. This paper also contains the conjecture, known in the literature as theLin–Ni conjecture and formulated in [36], that there are no nontrivial solutions for λ > 0 small enough and that there are non-trivial solutions forλlarge enough, even in the super-critical case p >2^∗−1. More details can be found in [37]. Partial results were obtained before in [41], when the exponent is in the range 1< p < d/(d−2). These papers were originally mo- tivated by the connection with the model of Keller and Segel in chemotaxis and the Gierer–Meinhardt system in pattern formation: see [40] for more explanations.

For completeness, let us briefly review what is known in the critical case p= 2^∗−1. Whend= 3, it was proved by M. Zhu in [50] that rigidity holds true forλ >0 small enough when one considers the positive solutions to the nonlinear elliptic equation

∆u−λ u+f(u) = 0

on a smooth bounded domain ofR³with homogeneous Neumann boundary conditions, iff(u) is equal tou⁵up to a perturbation of lower order. Another proof was given by J. Wei and X. Xu in [48] and slightly extended later in [30].

The Lin–Ni conjecture is wrong forp= 2^∗−1 in higher dimensions: see [47], and also earlier references therein. Some of the results have been extended to thed-Laplacian in dimension din [49].

Compared to the case of homogeneous Dirichlet boundary conditions or in the whole space, much less is known concerning bifurcations, qualita- tive aspects of the branches of solutions and multiplicity in the case of homogeneous Neumann boundary conditions. We may for instance refer to [38, 39, 43, 44, 45] for some results in this direction, but only in rather simple cases (balls, intervals or rectangles).

Concerning the Lin–Ni conjecture, it is known from [37, Theorem 3 (ii)]

that u ≡ λ^1/(p−1) if λ is small enough (also see [41] for an earlier partial result), and that there is a non-trivial solution ifλis large enough. As already said above, the method is based on the Moser iteration technique, in order to get a uniform estimate on the solution, and then on a direct estimate based on the Poincaré inequality. In the proof of Theorem 1.2 we shall adopt a completely different strategy, which is inspired by the rigidity results for nonlinear elliptic PDEs as in [7, 27] on the one hand, and by the carré du champ method of D. Bakry and M. Emery on the other hand, that can be traced back to [2]. More precisely, we shall rely on improved versions of these methods as in [4, 34], which involve the eigenvalues of the Laplacian, results on interpolation inequalities on compact manifolds obtained by J. Demange in [11], and a recent improvement with a computation based on traceless Hessians in [19].

From a larger perspective, our approach in Theorem 1.2 is based on re- sults for compact Riemannian manifolds that can be found in various papers:

the most important ones are the rigidity results of L. Véron et al. in [7, 33], the computations inspired by thecarré du champmethod of [4, 11], and the nonlinear flow approach of [19] (also see [14, 15, 18]). Using these estimates in the range 1< p <2^∗−1 and the Bakry–Emery method as in [25] in the casep∈(0,1), our goal is to prove that rigidity holds in a certain range ofλ without relying on uniform estimates (and the Moser iteration technique) and discuss the estimates of the threshold values. The spectral estimates of Theorem 1.1 are directly inspired by [16, 17].

This paper is organized as follows. Preliminary results have been collected in Section 2. The proof of Theorem 1.1 is given in Section 3. In Section 4, we use the heat flow to establish a first lower bound similar to the one of Theorem 1.2. Using a nonlinear flow a better bound is obtained in Section 5, which completes the proof of Theorem 1.2. The last section is devoted to various considerations on flows and, in particular, to improvements based on the nonlinear flow method.

Notation

IfA = (A_ij)_16i,j6d andB = (B_ij)_16i,j6d are two matrices, letA:B = Pd

i,j=1AijBij and|A|² =A:A. If aandb take values inR^d, we adopt the definitions:

a·b=

d

X

i=1

aibi, ∇ ·a=

d

X

i=1

∂ai

∂xi

,

a⊗b= (a_ib_j)_16i,j6d , ∇ ⊗a= ∂aj

∂x_i

16i,j6d

.

2. Preliminary results

Let us recall that Ω ⊂ R^d, d > 2 is a bounded domain with smooth boundary (or an open interval if d = 1) and let λ₂ be the first non-zero eigenvalue of the Laplace operator on Ω, supplemented with homogeneous Neumann boundary conditions. We shall denote byna unit outgoing normal vector of ∂Ω and will denote by u2 ∈ H¹(Ω) a non-trivial eigenfunction associated with the lowest positive eigenvalueλ2, so that

−∆u2=λ2u2 in Ω, ∂nu2= 0 on ∂Ω. (2.1) A trivial observation is thatu₂is in H²(Ω).

Lemma 2.1. — With the above notation, for any u ∈H²(Ω) such that

∂nu= 0 on∂Ω, we have Z

Ω

|∇u|²dx 2

6 Z

Ω

|∆u|²dx Z

Ω

|u|²dx . As a consequence, we also have

λ2

Z

Ω

|∇u|²dx6 Z

Ω

|∆u|²dx , (2.2)

and equality holds for any eigenfunction associated withλ₂.

Proof. — By expanding the square and integrating by parts the cross term, we notice that

06 Z

Ω

√1

µ∆u+λ2

õ u

2

dx

= 1

µk∆uk²_L2(Ω)−2λ2k∇uk²_L2(Ω)+λ²₂µkuk²_L2(Ω), whereµis an arbitrary positive real parameter. After optimizing on µ >0, we arrive at

062λ2

k∆uk_L2(Ω)kuk_L2(Ω)− k∇uk²_L2(Ω)

.

To check the equality case with u =u2, it is enough to multiply (2.1) by u2 and by−∆u2, and then integrate by parts. By definition ofλ2, we know

that Z

Ω

|∇u|²dx>λ₂ Z

Ω

|u|²dx if Z

Ω

udx= 0

with equality again ifu=u₂. This completes the proof. Notice indeed that the conditionR

Ωudx= 0 can always be imposed without loss of generality,

by adding the appropriate constant tou.

A key result for this paper is based on the computation of (∆u)²in terms of the Hessian matrix ofu, which involves integrations by parts and boundary terms. The following result can be found in [26, Lemma 5.1] or [28].

Lemma 2.2. — IfΩis a smooth convex domain inR^d and ifu∈C³(Ω) is such that∂_nu= 0 on∂Ω, then

−

d

X

i,j=1

Z

∂Ω

∂_ij²u ∂_iu n_jdH^d−1>0.

As a consequence, ifu∈H²(Ω) is such that∂_nu= 0, then we have that Z

Ω

|∆u|²dx>

Z

Ω

|Hessu|²dx .

In Lemma 2.2, the convexity is an essential ingredient, and this is where the convexity assumption comes from in all results of this paper.

Consider on H¹(Ω) the functional JΛ[u] :=k∇uk²_L2(Ω)− Λ

p−1

hkuk²_Lp+1(Ω)− kuk²_L2(Ω)

i

(2.3) ifp6= 1, and

JΛ[u] :=k∇uk²_L2(Ω)−Λ 2 Z

Ω

|u|² log |u|² kuk²_L2(Ω)

! dx ifp= 1.

Lemma 2.3. — There exists a function u ∈ H¹(Ω) such that ∂nu = 0 andJΛ[u]<0if Λ> λ2.

Proof. — A simple computations shows that J_Λ[1 + w]∼²h

k∇wk²_L2(Ω)−Λkwk²_L2(Ω)

i

as→0. By choosingw=u2 to be an eigenfunction associated withλ2, we get that

JΛ[1 + w]∼²(λ2−Λ)kwk²_L2(Ω)

is negative for >0 small enough if Λ> λ2. Lemma 2.3 provides the upper bound in Theorem 1.2. Indeed this proves that

Λ?6λ2.

This method has been widely exploited and a similar argument can be found for instance in [41].

3. Proof of Theorem 1.1

Assume thatp >1 and let us recall that µ(λ) := inf

u∈H¹(Ω)\{0}Qλ[u] with Qλ[u] := k∇uk²_L2(Ω)+λkuk²_L2(Ω)

kuk²_Lp+1(Ω)

. We denote byκp,dthe optimal constant in the following Gagliardo–Nirenberg inequality onR^d:

k∇vk²_L2(R^d)+kvk²_L2(R^d)>κ_p,dkvk²_Lp+1(R^d) ∀v∈H¹(R^d).

Lemma 3.1. — If p ∈(1,2^∗−1), the function λ 7→µ(λ) is monotone increasing, concave, such thatµ(λ)6λfor anyλ >0, andµ(λ) =λ if and only if0< λ6µ₂= Λ?/(p−1). Moreover, we have

µ(λ)∼2^1−p1+pκp,dλ^{1−d2 p−1p+1} as λ→+∞.

Proof. — For any givenu∈H¹(Ω)\ {0},λ7→ Qλ[u] is affine, increasing.

By taking an infimum, we know that, as a function ofλ,µis concave, non- decreasing. Usingu≡1 as a test function, we know thatµ(λ)6λfor any λ >0. By standard variational methods, we know that there is an optimal functionu∈H¹(Ω)\ {0}, so that

k∇uk²_L2(Ω)+λkuk²_L2(Ω)=µ(λ)kuk²_Lp+1(Ω). On the other hand, we know from (1.6) that

k∇uk²_L2(Ω)+ Λ?

p−1kuk²_L2(Ω)> Λ?

p−1kuk²_Lp+1(Ω). Hence we have the inequality

1− λ

Λ?

(p−1)

k∇uk²_L2(Ω)6(µ(λ)−λ)kuk²_Lp+1(Ω).

If λ 6 Λ?/(p−1), the l.h.s. is nonnegative while the r.h.s. is nonpositive because µ(λ) 6 λ, so that we conclude at once that µ(λ) = λ and u is constant. As a consequence,µ2>Λ?/(p−1). On the other hand, by definition of Λ?, we know thatµ26Λ?/(p−1).

The regime as λ→ ∞ is easily studied by a rescaling. If uλ denotes an optimal function such that Qλ[uλ] = µ(λ), then vλ(x) := uλ(x/√

λ) is an optimal function for

k∇vk²_L2(Ω_λ)+kvk²_L2(Ω_λ)> µ(λ) λ^{1−d2 p−1p+1}

kvk²_Lp+1(Ω_λ) ∀v∈H¹(Ωλ) where Ωλ := {x ∈ R^d : λ^−1/2x ∈ Ω}. Using truncations of the optimal functions for the Gagliardo–Nirenberg inequality onR^d+ ={(x1, x2, ...xd)∈ R^d : xd > 0} and an analysis of the convergence of an extension ofvλ in

H¹(R^d) as λ→ ∞ based on standard concentration-compactness methods, up to the extraction of subsequences and translations, we get that the limit functionv is optimal for the inequality

k∇vk²_L2(R^d₊)+kvk²_L2(R^d₊)>2^1−p1+pκ_p,dkvk²_Lp+1(R^d₊) ∀v∈H¹(R^d+). See [16, Lemma 5] for more details in a similar case.

By definition,λ7→ µ(λ) is monotone non-decreasing. As a consequence of the behavior at infinity and of the concavity property, this monotonicity is strict. Henceµis a monotone increasing function ofλ.

Assume thatp <1 and let us recall that λ(µ) := inf

u∈H¹(Ω)\{0}Q^µ[u] with Q^µ[u] := k∇uk²_L2(Ω)+µkuk²_Lp+1(Ω)

kuk²_L2(Ω)

. We denote byκ⁺_p,dthe optimal constant in the following Gagliardo–Nirenberg inequality onR^d+:

k∇vk²_L2(R^d₊)+kvk²_Lp+1(R^d₊)>κ⁺_p,dkvk²_L2(R^d₊) ∀v∈H¹(R^d+).

Lemma 3.2. — Ifp∈(0,1), the functionµ7→λ(µ)is monotone increas- ing, concave, such thatλ(µ)6µfor anyµ >0, andλ(µ) =µif and only if 0< µ6µ2= Λ?/(1−p). Moreover, we have

λ(µ)∼κ⁺_p,dµ ^{1+d2 1−pp+1 −1}

as µ→+∞.

Proof. — The proof follows the same strategy as in the proof of Lem- ma 3.1. See [16, Lemma 11] for more details in a similar case.

Recall that we denote byµ7→λ(µ) the inverse function ofλ7→µ(λ) and get in both cases,p >1 andp <1, the fact that

µ(λ) =O

λ^{1−d2 p−1p+1}

as λ→+∞.

Lemma 3.3. — Under the assumptions of Theorem 1.2, we haveν(µ) = λ(µ)for any µ >0 and, as a consequence,µ₂=µ₃.

Proof. — Assume first that p > 1. The proof is based on two ways of estimating the quantity

A=k∇uk²_L2(Ω)+λkuk²_L2(Ω)− Z

Ω

φ|u|²dx . On the one hand we may use Hölder’s inequality to estimate

Z

Ω

φ|u|²dx6kφk_Lq(Ω)kuk²_Lp+1(Ω)

withq= ^p+1_p−1 and get

A>k∇uk²_L2(Ω)+λkuk²_L2(Ω)−µkuk²_Lp+1(Ω)

withµ=kφkL^q(Ω). Usingu≡1 as a test function, we observe that the lowest eigenvalue λ1(Ω,−φ) of the Schrödinger operator −∆−φ is nonpositive.

Withu=u1 an eigenfunction associated withλ1(Ω,−φ), we know that A= (λ− |λ₁(Ω,−φ)|)kuk²_L2(Ω)>k∇uk²_L2(Ω)+λkuk²_L2(Ω)−µkuk²_Lp+1(Ω)

is nonnegative if λ= λ(µ), thus proving thatλ(µ)− |λ₁(Ω,−φ)| >0 and hence

λ(µ)>ν(µ).

On the other hand, withφ=µ u^p−1/kuk^p−1_Lp+1(Ω), we observe that 0 =A=k∇uk²_L2(Ω)+λkuk²_L2(Ω)−µkuk²_Lp+1(Ω)

>(λ− |λ1(Ω,−φ)|)kuk²_L2(Ω)>(λ−ν(µ))kuk²_L2(Ω)

if we takeµ=µ(λ) and uthe corresponding optimal function. This proves that

λ(µ)6ν(µ), which completes the proof whenp >1.

A similar computation can be done ifp <1, based on the Hölder inequal- ity

Z

Ω

u^p+1dx6 Z

Ω

u^p+1φ^p+12 φ^−p+12 dx6 Z

Ω

|u|²φdx ^p+1₂

kφ⁻¹k

p+1 2

L^q(Ω)

withq= ^1+p_1−p, that is Z

Ω

|u|²φdx>µkuk²_Lp+1(Ω)

withµ⁻¹=kφ⁻¹k_Lq(Ω). With

A=k∇uk²_L2(Ω)−λkuk²_L2(Ω)+ Z

Ω

φ|u|²dx ,

the computation is parallel to the one of the case p > 1. Also see [16] for

similar estimates.

Lemma 3.4. — Under the assumptions of Theorem 1.2, we haveµ₁6µ₂. Proof. — Letube an optimal function for (1.4). It can be taken nonneg- ative without restriction, and it solves the Euler–Lagrange equation

−ε(p) ∆u+λ u−µ u^p kuk^p−1_Lp+1(Ω)

= 0 in Ω, ∂nu= 0 on ∂Ω, (3.1)

where λ = λ(µ) or equivalently µ = µ(λ). By homogeneity, we can fix kukL^p+1(Ω) as we wish and may choose kuk^p−1_Lp+1(Ω) = µ, hence concluding thatuis constant ifµ6µ1 and, as a consequence,λ(µ) =µ, thus proving

thatµ6µ2. The conclusion follows.

4. Estimates based on the heat equation

We use the Bakry–Emery method to prove some results that are slightly weaker than the assertion of Theorem 1.2 but the method is of its own inde- pendent interest. Except for the precise value of the constant, the following result can be found in [25] (also see references to earlier papers therein).

Lemma 4.1. — Let d>1. Assume that Ωis a bounded convex domain such that |Ω|= 1. For anyp∈(0,1), for anyu∈H¹(Ω) such that ∂nu= 0 on∂Ω, we have

k∇uk²_L2(Ω)>λ2

hkuk²_L2(Ω)− kuk²_Lp+1(Ω)

i .

In this section and in the next section, we are going to use thecarré du champ method of D. Bakry and M. Emery in two different ways. Our goal is to prove that the functionalJ_Λ defined by (2.3) is nonnegative for some specific value of Λ>0.

•In theparabolic perspective, we will consider a flowt7→u(t,·) and prove

that d

dtJΛ[u(t,·)]6−β²R[u(t,·)]

for some non-zero parameter β and some nonnegative functional R. Since the flow drives the solutions towards constant functions, for whichJΛ takes the value 0, we henceforth deduce that

J_Λ[u(t,·)]> lim

s→+∞J_Λ[u(s,·)] = 0 ∀t>0.

As a consequence,JΛ[u0]>0 holds true for any initial datumu(t= 0,·) = u0∈H¹(Ω), which establishes the inequality. This approach has the advan- tage to provide for free a remainder term, since we know that

JΛ[u₀]>β² Z +∞

0

R[u(t,·)] dt .

The main disadvantage of the parabolic point of view is that it relies on the existence of a global and smooth enough solution. At least, this is compen- sated by the fact that one can take the initial datum as smooth as desired, prove the inequality and argue by density in H¹(Ω). Such issues are some- what standard and have been commented, for instance, in [46].

•Alternatively, one can adopt anelliptic perspective. Since, as we shall see, R[u] = 0 holds if and only ifuis constant on Ω, it is enough to consider an optimal function forJ_Λ, which is known to exist by standard compactness methods for any exponentpin the subcritical range, or even a positive critical point. The case of the critical exponent is more subtle, but can also be dealt with using techniques of the calculus of variations. An extremal functionu solves an Euler–Lagrange equation, which can be tested by a perturbation corresponding to the direction given by the flow. From a formal viewpoint, this amounts to take the solution to the flow problem with initial datumu0

and to compute _dt^dJΛ[u(t,·)] att= 0. However, no existence theory for the evolution equation is required and one can rely on the additional regularity properties that the functionu∈H¹(Ω) inherits as a solution to the Euler–

Lagrange equation. The elliptic regularity theoryà lade Giorgi–Nash–Moser is also somewhat standard but requires some care. The interested reader is invited to consult, for instance, [21] for further details on the application of this strategy, or [22] for a more heuristic introduction to the method.

In practice, we will use the two pictures without further notice. Detailed justifications and adaptations are left to the reader. Algebraically, in terms of integration by parts or tensor manipulations, the two methods are equivalent, and we shall focus on these computations, which explain why the method works but also underline its limitations.

Proof of Lemma 4.1. — We give a proof based on the entropy – entropy production method. It is enough to prove the result for nonnegative functions usince the inequality for|u|implies the inequality foru. By density, we may assume thatuis smooth. According to [25], ifv is a nonnegative solution of the heat equation

∂v

∂t = ∆v

on Ω with homogeneous Neumann boundary conditions, then v =u^p+1 is such that

d dt

Z

Ω

v^r−M^r

r−1 dx=−4 r

Z

Ω

|∇u|²dx withM :=R

Ωvdxandr= 2/(p+ 1). With this change of variables,usolves

∂u

∂t = ∆u+2−r r

|∇u|² u and we find that

− 1 2

d dt

Z

Ω

|∇u|²dx

= Z

Ω

|∆u|²dx+ p Z

Ω

|∇u|⁴

u² dx−2p Z

Ω

Hessu:∇u⊗ ∇u

u dx , (4.1)

that is,

− 1 2

d dt

Z

Ω

|∇u|²dx= 2r−1 r

Z

Ω

|∆u|²dx+p Z

Ω

|∆u|²− |Hessu|² dx +p

Z

Ω

Hessu−1

u∇u⊗ ∇u

2

dx , and finally, using Lemma 2.2,

d dt

Z

Ω

|∇u|²dx6−4r−1 r

Z

Ω

|∆u|²dx6−4r−1 r λ₂

Z

Ω

|∇u|²dx where the last inequality follows from Lemma 2.1, Ineq. (2.2), thus proving the result for anyp= (2−r)/r∈(0,1). Indeed, with previous notations, we have shown thatR

Ω|∇u|²dxis exponentially decaying. Hence Z

Ω

v^r−M^r

r−1 dx= 1 r−1

kuk²_L2(Ω)− kuk²_Lp+1(Ω)

also converges to 0 ast→ ∞and d

dt

k∇uk²_L2(Ω)−µ Z

Ω

v^r−M^r r−1 dx

6

−4r−1 r λ₂+4

rµ Z

Ω

|∇u|²dx is nonpositive ifµ6(r−1)λ2. Altogether, we have shown that

k∇uk²_L2(Ω)−λ2

kuk²_L2(Ω)− kuk²_Lp+1(Ω)

is nonincreasing with limit 0, which completes the proof.

If d > 2, a better result can be obtained by considering the traceless quantities as in [19]. Let us introduce

M[u] := ∇u⊗ ∇u

u −1

d

|∇u|²

u Id, (4.2)

Lu:= Hessu−1

d∆uId, (4.3)

and definep^]:= ^d_(d−1)^(d+2)2 so that

ϑ(p, d) :=p (d−1)² d(d+ 2)

satisfies ϑ(p, d) <1 for any p∈ (0, p^]). Notice that p^]+ 1 = ²_(d−1)^d2+12 is the threshold value that has been found in [3] (also see [14, 15]).

Lemma 4.2. — Let d>2. Assume that Ωis a bounded convex domain such that|Ω|= 1. For anyp∈(0, p^]), for anyu∈H¹(Ω)such that ∂nu= 0 on∂Ω, we have

k∇uk²_L2(Ω)>1

2 1−ϑ(p, d) λ2

kuk²_Lp+1(Ω)− kuk²_L2(Ω)

p−1

ifp6= 1and, in the limit case p= 1, k∇uk²_L2(Ω)> 1

4 1−ϑ(1, d) λ₂

Z

Ω

|u|² log |u|² kuk²_L2(Ω)

! dx .

The range ofpcovered in Lemma 4.2 is larger than the range covered in Lemma 4.1, but the constant is also better ifp∈ d(d+ 2)/(d²+ 6p−1),1 because, in that case, (1−ϑ(p, d))/(p−1)>2.

Proof. — We use the same conventions as in the proof of Lemma 4.1. Let us first observe that

|M[u]|²=

1−1 d

|∇u|⁴ u² ,

|Lu|²=|Hessu|²−1

d(∆u)².

Since Hessu= Lu+¹_d∆uId, we have that Hessu:∇u⊗ ∇u

u = Lu:∇u⊗ ∇u

u +1

d∆u|∇u|²

u = Lu: M[u] + 1

d∆u|∇u|² u

because Luis traceless. An integration by parts shows that Z

Ω

∆u|∇u|² u dx=

Z

Ω

|∇u|⁴ u² dx− 2

Z

Ω

Hessu:∇u⊗ ∇u

u dx

= d

d−1 Z

Ω

|M[u]|²dx−2 Z

Ω

Hessu:∇u⊗ ∇u

u dx

so that we get d+ 2

d Z

Ω

Hessu:∇u⊗ ∇u

u dx=

Z

Ω

Lu: M[u] dx+ 1 d−1

Z

Ω

|M[u]|²dx , hence

Z

Ω

Hessu:∇u⊗ ∇u

u dx

= d

d+ 2 Z

Ω

Lu: M[u] dx+ d (d−1) (d+ 2)

Z

Ω

|M[u]|²dx .

Now let us come back to the proof of Lemma 4.1. From (4.1), we read that

− 1 2

d dt

Z

Ω

|∇u|²dx

= Z

Ω

|∆u|²dx+p Z

Ω

|∇u|⁴

u² dx−2p Z

Ω

Hessu:∇u⊗ ∇u

u dx

= Z

Ω

|∆u|²dx+ p d d−1

Z

Ω

|M[u]|²dx

−2p d

d+ 2 Z

Ω

Lu: M[u] dx+ d (d−1) (d+ 2)

Z

Ω

|M[u]|²dx

= Z

Ω

|∆u|²dx−pd−1 d+ 2 Z

Ω

|Lu|²dx

+ p d²

(d−1) (d+ 2) Z

Ω

M[u]−d−1 d Lu

2

dx . We know from Lemma 2.2 that

Z

Ω

(∆u)²dx>

Z

Ω

|Hessu|²dx= Z

Ω

|Lu|²dx+1 d

Z

Ω

(∆u)²dx , i.e.,

Z

Ω

(∆u)²dx> d d−1

Z

Ω

|Lu|²dx . Altogether, this proves that, for anyθ∈(0,1),

−1 2

d dt

Z

Ω

|∇u|²dx>(1−θ) Z

Ω

|∆u|²dx+ θ d

d−1 −pd−1 d+ 2

Z

Ω

|Lu|²dx and finally, withθ=ϑ(p, d) and using (2.2),

d dt

k∇uk²_L2(Ω)−µ Z

Ω

v^r−M^r r−1 dx

6

−(1−θ)λ2+4 rµ

Z

Ω

|∇u|²dx is nonpositive if

µ6 r

4 1−ϑ(p, d)

λ₂= 1−ϑ(p, d) 2 (p+ 1) λ₂.

Since r−1 = (1−p)/(1 +p), this completes the proof if p6= 1. The case p= 1 is obtained by passing to the limit asp→1.

5. Estimates based on nonlinear diffusion equations

Lemma 5.1. — Assume that d>2 andΩ is a bounded convex domain inR^d with smooth boundary such that|Ω|= 1. Then we have

1−θ?(p, d)

|p−1| λ26µ1.

Proof. — This bound is inspired by [4, 19, 33, 34]. Let us give the main steps of the proof. Here we do it at the level of the nonlinear elliptic PDE.

Flows will be introduced afterwards, with the intent of providing improve- ments.

Let us consider the solutionuto (1.3) and define a functionv such that v^β =ufor some exponentβ to be chosen later. Thenv solves

−ε(p)

∆v+ (β−1)|∇v|² v

+λ v−v^κ= 0 in Ω (5.1) with homogeneous Neumann boundary conditions

∂nv= 0 on ∂Ω. (5.2)

Here

κ=β(p−1) + 1. (5.3)

If we multiply the equation by ∆v+κ|∇v|²/v

and integrate by parts, then the nonlinear term disappears and we are left with the identity

Z

Ω

(∆v)²dx+ (κ+β−1) Z

Ω

∆v|∇v|²

v dx+κ(β−1) Z

Ω

|∇v|⁴ v² dx

−λ|p−1|

Z

Ω

|∇v|²dx= 0. Using (4.2)–(4.3), let us define

Q[v] := Lv−(d−1) (p−1) θ(d+ 3−p) M[v].

The case of a compact manifold has been dealt with in [19]. The main dif- ference is that there is no Ricci curvature in case of a domain inR^d, but one has to take into account the boundary terms. As in the proof of Lemma 4.2, the main idea is to rearrange the various terms as a sum of squares of trace- less quantities. The computations for v are very similar to those done in Section 4, so we shall skip the details. The reader is invited to check that

θ Z

Ω

(∆v)²dx− Z

Ω

|Hessv|²dx

+ θ d d−1

Z

Ω

|Q[v]|²dx + (1−θ)

Z

Ω

(∆v)²dx−λ|p−1|

Z

Ω

|∇v|²dx= 0 if θ = θ_?(p, d) = _d^(d−1)_(d+2)+p^2p and β = _d+2−p^d+2 . In the previous identity, the first term is nonnegative by Lemma 2.2, the second term is the integral of a square and is therefore nonnegative, and the sum of the last ones is positive according to Lemma 2.1 if (1−θ)R

Ω(∆v)²dx−λ|p−1|R

Ω|∇v|²>0, unless

∇v= 0 a.e. Notice that the convexity of Ω is required to apply Lemma 2.2.

We may notice thatp=d+ 2 has to be excluded in order to defineβ, and this may occur ifd= 2. However, by working directly onu, it is possible to cover this case as well. This is indeed purely technical, because of the change of variablesu=v^β. Alternatively, it is enough to observe that the inequality holds for anyp6=d+ 2 and argue by continuity with respect top.

Proof of Theorem 1.2. — Since the exponent p is in the sub-critical range, it is classical that the functional JΛ has a minimizer u. Up to a normalizationv=u^1/β solves (5.1). If Λ =λ|p−1|<Λ?, thenuis constant by Lemma 5.1, and we are therefore in the caseλ=µ(λ) of Theorem 1.1 if λ6^1−θ_|p−1|^?(p,d)λ₂. Combined with the results of Theorem 1.1 and Lemma 2.3,

this completes the proof of Theorem 1.2.

For later purpose (see Section 6.3), let us consider the proof based on the flow. Withλ= Λ/|p−1|, we may consider the functionalu7→ JΛ[u] defined by (2.3) withu=v^β and evolve it according to

∂v

∂t =v^2−2β

∆v+κ|∇v|² v

. (5.4)

We also assume that (5.2) hold for anyt>0. This flow has the nice property

that d

dt Z

Ω

u^p+1dx= d dt

Z

Ω

v^β(p+1)dx= 0 ifκis given by (5.3), and a simple computation shows that

− 1 β²

d

dtJΛ[v^β] = Z

Ω

(∆v)²dx+ (κ+β−1) Z

Ω

∆v|∇v|² v dx +κ(β−1)

Z

Ω

|∇v|⁴

v² dx−λ|p−1|

Z

Ω

|∇v|²dx= 0. The same choices of β and θ as in the proof of Lemma 5.1 allow us to conclude, but it is interesting to discuss the possible values ofβ andθwhich guarantee that _dt^dJΛ[v^β]60 unless v is a constant. As in [19], elementary computations show that

− 1 β²

d dtJΛ[v^β]

=θ Z

Ω

(∆v)²dx− Z

Ω

|Hessv|²dx

+ θ d d−1

Z

Ω

|Q[v]|²dx +R

Z

Ω

|∇v|⁴

v² dx+ (1−θ) Z

Ω

(∆v)²dx−λ|p−1|

Z

Ω

|∇v|²dx where Q[u] is now defined by

Q[u] := Lu−1 θ

d−1

d+ 2(κ+β−1)

∇u⊗ ∇u

u −1

d

|∇u|² u Id

and

R:=−1 θ

d−1 d+ 2

²

(κ+β−1)²+κ(β−1) + (κ+β−1) d d+ 2. After replacingκby its value according to (5.3), we obtain that the equation

0 =R=

"

d−1 d+ 2

2

p²

θ −p+ 1

# β²−2

1− p

d+ 1

β+ 1 has two rootsβ_±(θ, p, d) ifθ∈ θ_?(p, d),1

andR >0 ifβ ∈(β₋, β₊). As in the linear case (proof of Lemma 4.2), we also know from Lemma 2.2 that

Z

Ω

(∆v)²dx− Z

Ω

|Hessv|²dx>0

and this is precisely where we take into account boundary terms and use the assumption that Ω is convex. Summarizing, we arrive at the following result.

Proposition 5.2. — With the above notation, if Ω is a bounded con- vex domain such that |Ω| = 1, for any θ ∈ θ_?(p, d),1

and any β ∈ β−(θ, p, d), β+(θ, p, d)

, we have d

dtJ_Λ[v^β]6−R β² Z

Ω

|∇v|⁴ v² dx ifv is a solution to (5.4).

When θ = θ_?(p, d), the reader is invited to check that β₋ = β₊ = β.

The computations in the proof of Lemma 5.1 can now be reinterpreted in the framework of the flow defined by (5.4). Up to the change of unknown functionu=v^β, any solution to (1.3) is stationary with respect to (5.4) and then all our computations amount to write that _dt^dJΛ[v^β] = 0 is possible only ifv is a constant.

6. Further considerations

Let us conclude this paper by a series of remarks. Section 6.1 is devoted to the question of the non-optimality in the lower bound of Theorem 1.2.

Spectral methods are introduced in Section 6.2 and provide us with an al- ternative method to establish (1.6) withp∈(0,1) when the constants in the extremal casesp= 0 (Poincaré inequality) andp= 1 (logarithmic Sobolev inequality) are known. The last estimates of Section 6.3 are based on refine- ments of the nonlinear flow method and extend the case of the manifolds with positive curvature studied in [15] to the setting of a bounded convex domain with homogeneous Neumann boundary conditions.